Gram-Schmidt Proof

I'm stuck on this one part for the proof of the Gram-Schmidt which states if is a linearly independent list (my text uses the term list, but I believe the term set is used more often) of vectors in V, then there exist an orthonormal list of vectors in V such that

for j=1,...,m

The proofs follows:

Suppose is a linearly independent list of vectors in V. To construct the e's, start by setting . This then satisfies the above equation.

So far in the proof I understand how it works for .

We will choose inductively, as follows. Suppose j>1 and an orthonormal list has been chosen so that

.

Here is where I am confused. I don't understand how an orthonormal list can be chosen such that the above is true. Isn't that similar to what we're trying to prove? And why can't the index j-1 just be j?

I did not post the complete proof; if this information is not sufficient enough I can post the rest.

I'm stuck on this one part for the proof of the Gram-Schmidt which states if is a linearly independent list (my text uses the term list, but I believe the term set is used more often) of vectors in V, then there exist an orthonormal list of vectors in V such that

for j=1,...,m

The proofs follows:

Suppose is a linearly independent list of vectors in V. To construct the e's, start by setting . This then satisfies the above equation.

So far in the proof I understand how it works for .

We will choose inductively, as follows. Suppose j>1 and an orthonormal list has been chosen so that

.

Here is where I am confused. I don't understand how an orthonormal list can be chosen such that the above is true. Isn't that similar to what we're trying to prove? And why can't the index j-1 just be j?

I did not post the complete proof; if this information is not sufficient enough I can post the rest.